Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D

TL;DR

This paper explores the capabilities of temperature 1 self-assembly models, demonstrating that 3D deterministic assembly and 2D probabilistic assembly can simulate complex systems and compute efficiently, contrasting with limitations in 2D deterministic assembly.

Contribution

The paper shows that temperature 1 self-assembly in 3D and probabilistic 2D models can simulate temperature 2 systems and perform efficient computations, overcoming previous limitations.

Findings

3D temperature 1 assembly can simulate temperature 2 systems.

Probabilistic 2D temperature 1 assembly can simulate Turing machines.

Near optimal $O( ext{log } n)$ tile complexity for assembling $n imes n$ squares.

Abstract

We investigate the power of the Wang tile self-assembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no temperature 1 assembly system has been shown to build a shape with a tile complexity smaller than the diameter of the shape. In contrast, we show that temperature 1 self-assembly in 3 dimensions, even when growth is restricted to at most 1 step into the third dimension, is capable of simulating a large class of temperature 2 systems, in turn permitting the simulation of arbitrary Turing machines and the assembly of $n\times n$ squares in near optimal $O(\log n)$ tile complexity. Further, we consider temperature 1 probabilistic assembly in 2D, and show that with a logarithmic scale up of tile complexity and shape scale, the same general class of temperature $\tau=2$ systems can be simulated with high probability, yielding Turing machine simulation and $O(\log^2 n)$ assembly of $n\times n$ squares with high probability. Our results show a sharp contrast in achievable tile complexity at temperature 1 if either growth into the third dimension or a small probability of error are permitted. Motivated by applications in nanotechnology and molecular computing, and the plausibility of implementing 3 dimensional self-assembly systems, our techniques may provide the needed power of temperature 2 systems, while at the same time avoiding the experimental challenges faced by those systems.